Problem
Finding equivalent fractions
Find the fractions equal to 3/7. An equal fraction multiplies top and bottom by the same whole number. The denominator must be two digits, from 10 to 99. Count how many there are.
Reasoning · Grade 3-1 Comparing Fraction Sizes
Your answer
Watch out Two-digit denominators are not 90 options — the denominator must be a multiple of 7.
How to solve
Strategy Look for a Pattern — Every fraction equal to 3/7 comes from multiplying both numerator and denominator by the same whole number k, so its denominator is 7 x k. We use that pattern to ask: for which k is 7 x k a two-digit number? Then we count those k values.
1STEP 1
Describe all fractions equal to 3/7
Every equal fraction has a denominator that is a multiple of 7.
3/7 = (3 × k)/(7 × k), denominator = 7k
Scaling the numerator and denominator by the same number is the basic rule for making equivalent fractions.
3.NF.A.3Look For A PatternMultiplying the top and the bottom by the same number gives a different-looking fraction that is the very same amount.
Why?
Cutting every piece into the same number of smaller pieces multiplies how many pieces you hold and how many make a whole by that same number.
🧱Scaling top and bottom keeps a fraction's valueMultiplying or dividing a fraction's top and bottom by the same number only renames it — the pieces change size but the amount you hold does not.
Why?
Nothing has actually been added or taken away, so the same share of the whole is being named a second way.
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
2STEP 2
Require the denominator to have two digits
The denominator has to sit between 10 and 99.
10 ≤ 7k ≤ 99
Turning the word 'two-digit' into the range 10 to 99 lets us test multiples of 7.
3.OA.A.4Look For A Pattern3STEP 3
Find the smallest and largest k
The smallest is 14 and the largest 98, since 105 has three digits.
7 × 2 = 14, 7 × 14 = 98, 7 × 15 = 105
Listing multiples of 7 shows exactly where they enter and leave the two-digit range.
3.OA.C.7Make A Systematic List4STEP 4
Count the values of k
The multiplier runs 2 to 14, so 13.
14 - 2 + 1 = 13
Counting how many whole numbers lie from 2 to 14 counts the fractions directly.
3.NBT.A.2Make A Systematic ListAnswer
13 fractions
14 − 2 + 1 = 13
Two-digit multiples of 7 are 14, 21, 28, ..., 98 — that is every multiple of 7 except 7 itself up to 98. There are 14 multiples of 7 up to 98, minus the one-digit 7, leaving 13, which matches.
Takeaway
Equivalent fractions of 3/7 just have denominator 7 x k, so you only count which times-7 facts land between 10 and 99!
- Describe all fractions equal to 3/7
- Require the denominator to have two digits
- Find the smallest and largest k
- Count the values of k